Boundary sets of regular and context-free languages

We investigate the descriptional and computational complexity of boundary sets of regular and context-free languages. For a letter a, the right (left, respectively) a-boundary set of a language L consists of the words in L whose a-predecessor or a-successor w.r.t. the prefix (suffix, respectively) relation is not in L. For regular languages described by deterministic finite automata (DFAs) we give tight bounds on the number of states for accepting boundary sets. Moreover, the question whether the boundary sets of a regular language is finite is shown to be NL -complete for DFAs, while it turns out to be PSPACE -complete for nondeterministic devices. Boundary sets for context-free languages are not necessarily context free anymore. Here we find a subtle difference of right and left a-boundary sets. While right a-boundary sets of deterministic context-free languages stay deterministic context free, we give an example of a deterministic context-free language whose a-boundary set is already non-context free. In fact, the finiteness problem for a-boundary sets of context-free languages becomes undecidable.

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